Algorithmic Cost Allocation Games: Theory and Applications
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Algorithmic Cost Allocation Games: Theory and Applications vorgelegt von Dipl.-Math. Nam D˜ung Ho`ang aus Hanoi, Vietnam Von der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss Vorsitzender: Prof. Dr. Fredi Tr¨oltzsch Berichter: Prof. Dr. Dr. h.c. mult. Martin Gr¨otschel Dr. habil. Ralf Bornd¨orfer Tag der wissenschaftlichen Aussprache: 6. Oktober 2010 Berlin 2010 D 83 For my parents Abstract Due to economy of scale, it is suggested that individual users, in order to save costs, should join a cooperation rather than acting on their own. However, a challenge for individuals when cooperating with others is that every member of the cooperation has to agree on how to allocate the com- mon costs among members, otherwise the cooperation cannot be realised. Taken this issue into account, we set the objective of our thesis in inves- tigating the issue of fair allocations of common costs among users in a cooperation. This thesis combines cooperative game theory and state-of- the-art algorithms from linear and integer programming in order to define fair cost allocations and calculate them numerically for large real-world ap- plications. Our approaches outclasse traditional cost allocation methods in terms of fairness and users’ satisfaction. Cooperative game theory analyzes the possible grouping of individuals to form their coalitions. It provides mathematical tools to understand fair prices in the sense that a fair price prevents the collapse of the grand coali- tion and increases the stability of the cooperation. The current definition of cost allocation game does not allow us to restrict the set of possible coalitions of players and to set conditions on the output prices, which of- ten occur in real-world applications. Our generalization bring the cost allocation game model a step closer to practice. Based on our definition, we present and discuss in the thesis several mathematical concepts, which model fairness. This thesis also considers the question of whether there exists a “best” cost allocation, which people naturally like to have. It is well-known that multicriteria optimization problems often do not have “the optimal solu- tion” that simultaneously optimizes each objective to its optimal value. There is also no “perfect” voting-system which can satisfy all the five sim- ple, essential social choice procedures presented in the book “Mathematics and Politics. Strategy, Voting, Power and Proof” of Taylor et al. Similarly, the cost allocation problem is shown to experience the same problem. In i particular, there is no cost allocation which can satisfy all of our desired properties, which are coherent and seem to be reasonable or even indis- pensable. Our game theoretical concepts try to minimize the degree of axiomatic violation while the validity of some most important properties is kept. From the complexity point of view, it is NP-hard to calculate the allo- cations which are based on the considered game theoretical concepts. The hardest challenge is that we must take into account the exponential num- ber of the possible coalitions. However, this difficulty can be overcome by using constraint generation approaches. Several primal and dual heuris- tics are constructed in order to decrease the solving time of the separation problem. Based on these techniques, we are able to solve our applications, whose sizes vary from small with 4 players, to medium with 18 players, and even large with 85 players and 285 − 1 possible coalitions. Via computa- tional results, we show the unfairness of traditional cost allocations. For example, for the ticket pricing problem of the Dutch IC railway network, the current distance tariff results in a situation where the passengers in the central Randstad region of the country pay over 25% more than the costs they incur and these excess payments subsidize operations elsewhere, which is absolutely not fair. In contrast, our game theory based prices decrease this unfairness and increase the incentive to stay in the grand coalition for players. ii Acknowledgments I would like to express my sincere gratitude to my advisor Prof. Dr. Dr. h.c. mult. Martin Gr¨otschel for the interesting research theme and for his supervision. I am grateful to Dr. Ralf Bornd¨orfer for his valuable supports and suggestions. I would like to express my thank to the Zuse Institute Berlin (ZIB) for providing me a Konrad-Zuse Scholarship. I also want to thank to all my friends and colleagues at ZIB for the wonderful working atmosphere. Moreover, I am appreciative to my proof readers Carlos Cardonha, Dr. Benjamin Hiller, and Dr. Thorsten Koch for their precious comments. And last but not least, I would like to thank my parents for their care and continual supports. iii Contents Abstract i Acknowledgments iii 1 Introduction 1 2 TheCostAllocationProblem 9 2.1 TheCostAllocationGame. 11 2.2 DesiredPropertiesandConflicts. 13 2.2.1 DesiredProperties . 14 2.2.2 Conflicts......................... 18 2.2.3 SomeOtherDesiredProperties . 21 2.3 GameTheoreticalConcepts . 28 2.3.1 The Core and the f-LeastCore . 28 2.3.2 The f-Nucleolus..................... 51 2.3.3 The (f,r)-LeastCore.................. 66 2.3.4 ChoosingtheWeightFunction . 72 2.3.5 TheShapleyValue . 73 2.3.6 AnotherConflict . 77 2.4 AlternativeAnsatz ....................... 78 2.5 Non-emptinessoftheCore . 78 2.6 Conclusions ........................... 87 3 Complexity 91 3.1 NP-HardnessofCostAllocationGame . 91 3.2 Ellipsoid Method and Submodular Function Minimization . 92 3.3 Polynomial Time Algorithms for Submodular Games . 94 3.3.1 Algorithm for the f-Nucleolus . 95 v CONTENTS 3.3.2 Algorithms for the f-Least Core and the (f,r)-Least Core ...........................106 4 Computational Aspects 107 4.1 CombinatorialGame . .108 4.2 ConstraintsGenerationApproaches . 108 4.2.1 The f-Least Core and the f-Nucleolus . .108 4.2.2 The (f,r)-LeastCore. .110 4.2.3 ChoosingAGoodStartingSet. 112 4.2.4 TheSeparationProblem . .121 4.2.5 Heuristics for the Separation Problem . 123 5 TheFairnessDistributionDiagram 127 6 A Simple Real Example 131 7 Allocating Production and Transmission Costs 137 7.1 ProductionandTransmissionCosts . 137 7.2 Nonlinear Multi-commodity Flow Model . 138 7.3 MixedIntegerModel . .140 7.4 PiecewiseLinearApproximation . 148 7.5 Cost Allocation in Water Resources Development . 154 7.5.1 The Water Resources Development Cost Allocation Game ..........................155 7.5.2 TheSeparationProblem . .160 7.5.3 ComputationalResults . 162 8 TicketPricingProbleminPublicTransport 167 8.1 TheTicketPricingProblem . .167 8.2 CalculatingtheCostFunction . 170 8.3 TheSeparationProblem . .173 8.4 TicketPricesfortheDutchICNetwork. 175 9 Perspectives 183 Bibliography 185 Notations 193 Index 195 vi Chapter 1 Introduction There has been an endless controversy on a fair price system for train slots in Germany for years. The current one of the German railway infrastruc- ture provider DB Netz AG is accused of being incommensurate with the real cost. The train path charges per train path kilometer are composed of route category, train path product, service-dependent component, regional factor, and several other components. DB Netz AG has subdivided its routes in 12 categories based on the maximal allowed speed, which reflects the investment cost. The higher the maximal allowed speed of a route, the more expensive is the basic price for one transport kilometer on this route. There are 9 different train path products with 5 products for passenger transport and 4 products for freight transport. Each of them is assigned to a factor from 0.5 to 1.8. This classification is based on the customer’s demands, e.g., direct connection, priority in terms of operations manage- ment, and/or frequency. DB Netz AG claims that the role of the service- dependent component is to provide “an incentive to reduce disturbances and improve the efficiency of the rail network” by applying some penalty factors on “very busy routes” and for trains where a minimum speed of 50 km/h is not achieved. As revealed from its name, regional factor differs locally depending on the regional network concerned. They represent a supplement on the top of the train path price. The train path price is then the product of these components. However, it is unclear how DB Netz AG came to these numbers. In the article “Schienennetz wird f¨ur Deutsche Bahn zum blendenden Gesch¨aft” from 18.07.2009 in Wirtschaftswoche, a German weekly business news magazine, Christian Schlesiger wrote that: The path prices are often incommensurate to the costs caused by the trains. Example Hamburg-Berlin: On this route local 1 1. Introduction and long distance trains pay the same amount of 6.95 Euros per kilometer, although the load on the rails and railroad switches is different. The ICE requires the triple of the capacity of the regional trains, needs numerous expensive extra facilities, a spe- cial trolley, and high-speed switches. According to a calculation of KCW, a management consulting firm in public transport, the charges of ICE ought to be 10.43 Euros per kilometer, while a regional train would pay only 3.48 Euros. For Deutsche Bahn that would be very unattractive, since it would have to trans- port 37 passengers more per ICE in order to obtain the same profit. The “Interessengemeinschaft” Bibertbahn, which has striven for years for the reactivation of the railway line Nuremberg-Stein-Altenberg-Zirndorf- Leichendorf, also complained on the high path price for the 5.6 kilometer route between Stein and the Nuremberg main station. DB Netz AG asked for about one million euros per year, while the community claimed that according to its estimation 200 thousands euros should be a reasonable price since the extra cost for DB Netz AG is very small. These are only two of many complaints on the DB Netz AG price system.